Advanced calculus
Free

Advanced calculus

By Lynn H. Loomis
Free
Book Description
Table of Contents
  • Title Page
  • Preface
  • Contents
  • Chapter 0: Introduction
    • 0.1 Logic: Quantifiers
    • 0.2 The Logical Connectives
    • 0.3 Negations of Quantifiers
    • 0.4 Sets
    • 0.5 Restricted Variables
    • 0.6 Ordered Pairs and Relations
    • 0.7 Functions and Mappings
    • 0.8 Product Sets; Index Notation
    • 0.9 Composition
    • 0.10 Duality
    • 0.11 The Boolean Operations
    • 0.12 Partitions and Equivalence Relations
  • Chapter 1: Vector Spaces
    • 1.1 Fundamental Notions
    • 1.2 Vector Spaces and Geometry
    • 1.3 Product Spaces and Hom(V,W)
    • 1.4 Affine Subspaces and Quotient Spaces
    • 1.5 Direct Sums
    • 1.6 Bilinearity
  • Chapter 2: Finite-Dimensional Vector Spaces
    • 2.1 Bases
    • 2.2 Dimension
    • 2.3 The Dual Space
    • 2.4 Matrices
    • 2.5 Trace and Determinant
    • 2.6 Matrix Computations
    • 2.7* The Diagonalization of a Quadratic Form
  • Chapter 3: The Differential Calculus
    • 3.1 Review in R
    • 3.2 Norms
    • 3.3 Continuity
    • 3.4 Equivalent Norms
    • 3.5 Infinitesimals
    • 3.6 The Differential
    • 3.7 Directional Derivatives; the Mean-Value Theorem
    • 3.8 The Differential and Product Spaces
    • 3.9 The Differential and R^n
    • 3.10 Elementary Applications
    • 3.11 The Implicit-Function Theorem
    • 3.12 Submanifolds and Lagrange Multipliers
    • 3.13* Functional Dependence
    • 3.14* Uniform Continuity and Function-Valued Mappings
    • 3.15* The Calculus of Variations
    • 3.16* The Second Differential and the Classification of Critical Points
    • 3.17* Higher Order Differentials. The Taylor Formula
  • Chapter 4: Compactness and Completeness
    • 4.1 Metric Spaces; Open and Closed Sets
    • 4.2* Topology
    • 4.3 Sequential Convergence
    • 4.4 Sequential Compactness
    • 4.5 Compactness and Uniformity
    • 4.6 Equicontinuity
    • 4.7 Completeness
    • 4.8 A First Look at Banach Algebras
    • 4.9 The Contraction Mapping Fixed-Point Theorem
    • 4.10 The Integral of a Parameterized Arc
    • 4.11 The Complex Number System
    • 4.12* Weak Methods
  • Chapter 5: Scalar Product Spaces
    • 5.1 Scalar Products
    • 5.2 Orthogonal Projection
    • 5.3 Self-Adjoint Transformations
    • 5.4 Orthogonal Transformations
    • 5.5 Compact Transformations
  • Chapter 6: Differential Equations
    • 6.1 The Fundamental Theorem
    • 6.2 Differentiable Dependence on Parameters
    • 6.3 The Linear Equation
    • 6.4 The nth-Order Linear Equation
    • 6.5 Solving the Inhomogeneous Equation
    • 6.6 The Boundary-Value Problem
    • 6.7 Fourier Series
  • Chapter 7: Multilinear Functionals
    • 7.1 Bilinear Functionals
    • 7.2 Multilinear Functionals
    • 7.3 Permutations
    • 7.4 The Sign of a Permutation
    • 7.5 The Subspace A^n of Alternating Tensors
    • 7.6 The Determinant
    • 7.7 The Exterior Algebra
    • 7.8 Exterior Powers of Scalar Product Spaces
    • 7.9 The Star Operator
  • Chapter 8: Integration
    • 8.1 Introduction
    • 8.2 Axioms
    • 8.3 Rectangles and Paved Sets
    • 8.4 The Minimal Theory
    • 8.5 The Minimal Theory (Continued)
    • 8.6 Contented Sets
    • 8.7 When is a Set Contented?
    • 8.8 Behavior Under Linear Distortions
    • 8.9 Axioms for Integration
    • 8.10 Integration of Contented Functions
    • 8.11 The Change of Variables Formula
    • 8.12 Successive Integration
    • 8.13 Absolutely Integrable Functions
    • 8.14 Problem Set: The Fourier Transform
  • Chapter 9: Differentiable Manifolds
    • 9.1 Atlases
    • 9.2 Functions, Convergence
    • 9.3 Differentiable Manifolds
    • 9.4 The Tangent Space
    • 9.5 Flows and Vector Fields
    • 9.6 Lie Derivatives
    • 9.7 Linear Differential Forms
    • 9.8 Computations with Coordinates
    • 9.9 Riemann Metrics
  • Chapter 10: The Integral Calculus on Manifolds
    • 10.1 Compactness
    • 10.2 Partitions of Unity
    • 10.3 Densities
    • 10.4 Volume Density of a Riemann Metric
    • 10.5 Pullback and Lie Derivatives of Densities
    • 10.6 The Divergence Theorem
    • 10.7 More Complicated Domains
  • Chapter 11: Exterior Calculus
    • 11.1 Exterior Differential Forms
    • 11.2 Oriented Manifolds and the Integration of Exterior Differential Forms
    • 11.3 The Operator d
    • 11.4 Stokes' Theorem
    • 11.5 Some Illustrations of Stokes' Theorem
    • 11.6 The Lie Derivative of a Differential Form
    • Appendix I: "Vector Analysis"
    • Appendix II: Elementary Differential Geometry of Surfaces in E^3
  • Chapter 12: Potential Theory in E^n
    • 12.1 Solid Angle
    • 12.2 Green's Formulas
    • 12.3 The Maximum Principle
    • 12.4 Green's Functions
    • 12.5 The Poisson Integral Formula
    • 12.6 Consquences of the Poisson Integral Formula
    • 12.7 Harnack's Theorem
    • 12.8 Subharmonic Functions
    • 12.9 Dirichlet's Problem
    • 12.10 Behavior Near the Boundary
    • 12.11 Dirchlet's Principle
    • 12.12 Physical Applications
    • 12.13 Problem Set: The Calculus of Residues
  • Chapter 13: Classical Mechanics
    • 13.1 The Tangent and Cotangent Bundles
    • 13.2 Equations of Variation
    • 13.3 The Fundamental Linear Differential Form on T*(M)
    • 13.4 The Fundamental Exterior Two-Form on T*(M)
    • 13.5 Hamiltonian Mechanics
    • 13.6 The Central-Force Problem
    • 13.7 The Two-Body Problem
    • 13.8 Lagrange's Equations
    • 13.9 Variational Principles
    • 13.10 Geodesic Coordinates
    • 13.11 Euler's Equations
    • 13.12 Rigid Body Motion
    • 13.13 Small Oscillations
    • 13.14 Small Osccillations (Continued)
    • 13.15 Canonical Transformations
  • Selected References
  • Notation Index
  • Index
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